MDP-Based Recommender System

• MDP-Based Recommender System is a framework that models recommendations as a sequential decision-making process, integrating states, actions, transitions, rewards, and discounting.
• It employs compact state representations and RL algorithms such as Q-learning, PPO, and actor-critic methods to address challenges like cold-start and scalability.
• Empirical studies demonstrate improved recommendation accuracy, explainability, and performance across diverse datasets like MovieLens and Amazon.

A Markov Decision Process (MDP)-based recommender system formalizes recommendation as a sequential decision-making problem, enabling optimization of long-term user engagement, satisfaction, or business objectives. The system models the user-system interaction as an MDP, which consists of states, actions, transitions, rewards, and a discount factor, and deploys reinforcement learning (RL) algorithms to learn policies that maximize expected cumulative reward over time. This paradigm allows recommender systems to take into account the future consequences of current recommendations and systematically balance short-term and long-term objectives.

1. Formal MDP Modeling of Recommendation

An MDP for recommendation is defined as the tuple $(\mathcal{S},\mathcal{A},P,R,\gamma)$, where:

• State space ($\mathcal{S}$): Encodes the user's recent interaction history or latent preferences. States can be tabular (each item or history window as a symbol), embedding-based (vector representations from static features), or sequence-modeled (RNNs or Transformers over item sequences) (Afsar et al., 2021, Shani et al., 2012, Zhao et al., 2017, Lu et al., 2016, Wang et al., 23 Jan 2025, Bai et al., 2019, Choi et al., 2018, Chen et al., 2022).
• Action space ($\mathcal{A}$): Items or slates (bundles) which the system can recommend at each step. Actions may involve single-item recommendation, slate/bundle selection, or grid-based slot allocation (Afsar et al., 2021, Zhao et al., 2017, Chen et al., 2022).
• Transition kernel ($P$): Probability distribution over next states given current state-action. Often estimated from logs or simulated via learned user models (e.g., n-gram predictors, RNNs, GAN-based simulators) (Shani et al., 2012, Bai et al., 2019).
• Reward ($R$): Immediate numerical feedback signaling utility of an action (click, purchase, rating, dwell time, etc.), or shaped composite functions for long-term planning (Afsar et al., 2021, Wang et al., 23 Jan 2025, Chen et al., 2022, Lu et al., 2016).
• Discount factor ($\gamma$): Governs tradeoff between immediate and future rewards.

MDP optimization seeks a stationary policy $\pi^*$ maximizing the expected discounted return,

$$J(\pi)=\mathbb{E}\Bigl[\sum_{t=0}^\infty \gamma^t R(s_t, a_t) \Bigr].$$

2. State Representation and Dimensionality Reduction

The practical deployment of MDP-based recommenders hinges on compact yet expressive state representations:

• Tabular and history-windowed states: Early work encodes states as fixed-length histories (e.g., $k$ most recent selections), resulting in $O(|I|^k)$ state complexity (Shani et al., 2012). State pruning and mixture modeling alleviate combinatorial explosion.
• Embedding and latent belief states: Matrix factorization and embedding methods represent user/item features in continuous space, enabling efficient belief tracking (POMDP-Rec collapses the full posterior to a MAP embedding) (Lu et al., 2016, Afsar et al., 2021, Bai et al., 2019).
• Biclustering for gridworld abstraction: Biclustering the user-item matrix produces dense submatrices $(U_s, I_s)$, mapped to $n^2$ grid states, massively reducing dimensionality and action space. Movement in the gridworld directly corresponds to transitions between semantically meaningful biclusters (Choi et al., 2018).
• Sequence encoders: Modern architectures use RNNs or Transformers to encode dynamic interaction histories into continuous vectors, supporting high-fidelity estimation of user state (Wang et al., 23 Jan 2025, Bai et al., 2019, Zhao et al., 2017).

These compact representations facilitate scalable RL policy optimization, even with large user/item catalogs.

3. Reinforcement Learning Algorithms and Solution Methods

MDP-based recommenders deploy a range of RL algorithms, tailored to the complexity of the state and action spaces:

• Tabular algorithms: Q-learning, SARSA on reduced gridworlds (small $|S|$ and $|A|$ via biclustering) (Choi et al., 2018).
• Actor-critic and policy gradient methods: Advantage Actor-Critic (A2C), Proximal Policy Optimization (PPO), Deep Deterministic Policy Gradient (DDPG), used for high-dimensional and continuous action spaces (Zhao et al., 2017, Wang et al., 23 Jan 2025, Chen et al., 2022, Afsar et al., 2021).
• Model-based RL: Learned user models via n-gram chains, RNNs, or GANs serve as simulators for offline policy evaluation and planning. IRecGAN couples an adversarial discriminator with the agent-environment generator to minimize bias and optimize policy using discriminator-weighted rewards (Shani et al., 2012, Bai et al., 2019).
• Partial Observability: POMDP-Rec maintains belief distributions over hidden user states, updating via observed feedback; fitted-Q over belief transitions is solved with neural networks (Lu et al., 2016).
• Exploration/Exploitation: Policies employ $\epsilon$-greedy, Boltzmann, categorical, or adaptive mixing (policy combination with annealing) to balance reward maximization and environment exploration (Choi et al., 2018, Wang et al., 23 Jan 2025).

Distributed training, prioritized replay, and experience simulators further boost sample efficiency.

4. Reward Design and Evaluation Metrics

Reward functions instantiate the system’s objectives and are central to both learning and offline/online evaluation:

• Instantaneous rewards: Direct outcomes such as clicks, purchases, or ratings (Afsar et al., 2021, Zhao et al., 2017, Chen et al., 2022, Wang et al., 23 Jan 2025).
• Delayed/long-term rewards: Metrics that reflect sustained engagement, such as session length, repeat visits, dwell time.
• Reward shaping: Combines multiple signals through weighted sums for composite optimization (Afsar et al., 2021).
• LLM-driven distillation: Offline pretraining via LLMs by prompting for preferences, setting synthetic binary rewards (Wang et al., 23 Jan 2025).
• Adversarial scaling: IRecGAN scales simulated rewards by discriminator output to reduce model bias and off-policy errors (Bai et al., 2019).

Empirical evaluation uses precision@N, recall@N, MAP, NDCG, coverage@r, RMSE, and session-level returns. Statistical analysis (e.g., $p<0.05$ in Panel-MDP) asserts significant improvements over baselines.

5. Practical Deployment, Scalability, and Environment Simulation

MDP-based recommenders face production challenges arising from state/action space explosion, cold start, distributional shift, and safe exploration:

• State/action reduction: Biclustering, grid mapping, mixture modeling, and slate recommendations allow tractable RL in large environments (Choi et al., 2018, Zhao et al., 2017, Chen et al., 2022, Afsar et al., 2021).
• Simulators: Offline training relies on simulators built from historical logs (n-gram chains, collaborative filtering models, RNN-based behavior models, GAN user–agent generators) to roll out trajectories and pretrain policies (Shani et al., 2012, Bai et al., 2019, Zhao et al., 2017, Wang et al., 23 Jan 2025).
• Cold start: Latent factor models, biclustering, and LLM-driven preference extraction mitigate the lack of historical data for new users/items (Choi et al., 2018, Wang et al., 23 Jan 2025).
• Online adaptation: Post-deployment schemes (A-iALP$_{ft}$ and adaptive mixing A-iALP$_{ap}$) mitigate distribution shift and unstable policies by fine-tuning or gradually mixing new policies (Wang et al., 23 Jan 2025).
• Panel/grid-based paneling: Panel-MDP extends MDP modeling to grid-style layouts by sequentially deciding exposure and slot allocation, supporting complex layout optimization (Chen et al., 2022).

Systematic evaluation on industry-scale datasets (JD.com, MovieLens, CIKM Cup, Yahoo Music, Amazon) confirms scalability and real-world impact.

6. Explainability, Emerging Trends, and Limitations

MDP-based recommenders facilitate interpretability and the integration of advanced sequential modeling:

• Explainability: State-based explanations are supported by biclustering, as recommendations are justified by user–item cluster membership (Choi et al., 2018).
• Sequential, listwise, gridwise optimization: Modern methods explicitly model multi-item recommendations, bundle diversity, and spatial allocation (e.g., listwise and grid-panel approaches) (Zhao et al., 2017, Chen et al., 2022).
• Partial observability: POMDP-Rec formalizes recommender interaction under unobserved user states, optimizing long-term accuracy without negative sampling (Lu et al., 2016).
• Adversarial model-based learning: GAN-based frameworks address offline evaluation bias and policy debiasing (Bai et al., 2019).

Challenges persist in scaling to extreme catalog sizes, combating recurrent deterioration from bias and interest drift, ensuring robust offline evaluation, and improving explainability in deep models. Recent advances in LLM-driven preference extraction and adaptive online policy mixing show marked improvements in cold-start and long-run return (Wang et al., 23 Jan 2025).

7. Representative Results and Comparative Performance

Across benchmarks, MDP-based recommender systems yield superior performance in both cold-start and long-session regimes:

Approach	Dataset	Precision@30	Recall@30	MAP/NDCG/Other	Key Finding
RL+Biclustering	ML 100K/1M	0.246/0.277	0.169/0.155	—	Outperforms CF in cold-start (Choi et al., 2018)
Listwise DDPG (LIRD)	JD.com e-commerce	—	—	best MAP/NDCG at K=4	Trains faster and scales better than vanilla DQN (Zhao et al., 2017)
POMDP-Rec	MovieLens, Yahoo Music	—	—	RMSE 0.8419/22.769	No need for negatives, stable under reiteration (Lu et al., 2016)
IRecGAN	CIKM Cup, simulated	—	—	P@10=35.06%, cov@r	Superior sample efficiency and reward over PG/AC baselines (Bai et al., 2019)
Panel-MDP (PPO)	E-commerce grids	—	—	reward 0.0179*, AUC 0.7980*	Grid allocation with PPO outperforms DDPG, miDNN (Chen et al., 2022)
LLM+A2C (A-iALP)	LFM, Amazon, Coat	—	—	R@1=8.83/9.28	Cold-start and online adaptation gains, fast convergence (Wang et al., 23 Jan 2025)

A plausible implication is that the MDP framing and associated RL algorithms provide an essential infrastructure for next-generation recommendation systems, supporting explainability, scalability, and optimal sequential decision-making under partial observability and long-term constraints.